The spectrum of the curl operator on spherically symmetric domains

This paper presents a mathematically complete derivation of the minimum-energy divergence-free vector fields of fixed helicity, defined on and tangent to the boundary of solid balls and spherical shells. These fields satisfy the equation ∇×V=λV, where λ is the eigenvalue of curl having smallest nonzero absolute value among such fields. It is shown that on the ball the energy minimizers are the axially symmetric spheromak fields found by Woltjer and Chandrasekhar–Kendall, and on spherical shells they are spheromak-like fields. The geometry and topology of these minimum-energy fields, as well as of some higher-energy eigenfields, are illustrated.